Abstract:

The Sierpinski carpet fractal is one best known planar, compact, connected sets. It is a universal plane continuum and there is a topological characterization of this set. Any planar set homeomorphic to a Sierpinski carpet is called a Sierpinski curve. In recent years several authors have shown that Sierpinski curve can arise as the Julia set of certain holomorphic functions. We make an attempt towards a more systematic approach to the problem of existence of Sierpinski curves as Julia sets of rational maps. Our goal is to find dynamical conditions under which we can assure that the Julia set of a certain quadratic rational map is a Sierpinski curve.